IAT (IISER) 2023 | Motion in a Straight Line Question 2 | Physics

$$ v(t)= \begin{cases}\alpha t, & 0 \leq t \leq T / 3 \\ \alpha T / 3, & T / 3 \leq t \leq 2 T / 3 \\ \alpha(T-t), & 2 T / 3 \leq t \leq T\end{cases} $$

where $\alpha(\neq 0)$ is a constant. What is the displacement of the particle from time $t=0$ to $T$ ?

$\frac{2 \alpha T^2}{9}$

$\frac{4 \alpha T^2}{9}$

$\frac{8 \alpha T^2}{9}$

$\frac{7 \alpha T^2}{9}$

IAT (IISER) 2022

MCQ (Single Correct Answer)

-1

A particle experiences an acceleration $\vec{a}=\alpha \vec{v}$, where $\vec{v}$ is the velocity of the particle and $\alpha$ is a constant. If the distances traveled by the particle in the time intervals $t_2-t_1$ and $t_3-t_1$ are $S_{12}$ and $S_{13}$, respectively, which of the following relations is true?

$ \frac{S_{13}}{S_{12}}=\frac{\log \left[\alpha\left(t_3-t_1\right)\right]}{\log \left[\alpha\left(t_2-t_1\right)\right]}$

$\frac{S_{13}}{S_{12}}=\frac{\exp \left[\alpha\left(t_3-t_1\right)\right]}{\exp \left[\alpha\left(t_2-t_1\right)\right]}$

$\frac{S_{13}}{S_{12}}=\frac{\exp \left[\alpha\left(t_3-t_1\right)\right]-1}{\exp \left[\alpha\left(t_2-t_1\right)\right]-1}$

$\frac{S_{13}}{S_{12}}=\frac{\log \left[\alpha\left(t_3-t_1\right)\right]-1}{\log \left[\alpha\left(t_2-t_1\right)\right]-1}$

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